Last-Passage American Cancelable Option in Lévy Models

We derive the explicit price of perpetual American put option canceled at last-passage time underlying above some fixed level. assume that asset process is governed by a geometric spectrally negative Lévy process. show optimal exercise first moment when drops below an threshold. perform numerical analysis considering classical Black–Scholes models and model where logarithm has additional exponential downward shocks. The proof based on martingale arguments fluctuation theory processes.